Global universal approximation of functional input maps on weighted spaces

Understand

We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space.

• To this end, we use an additive family to map the input weighted space to the hidden layer, on which a non-linear scalar activation function is applied to each neuron, and finally return the output via some linear readouts.
• Relying on Stone-Weierstrass theorems on weighted spaces, we can prove a global universal approximation result on weighted spaces for continuous functions going beyond the usual approximation on compact sets.
• This then applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks.